MA102: Single-Variable Calculus II

Unit 2: Applications of IntegrationIn this unit, we will take a first look at how integration can and has
been used to solve various types of problems. Now that you have
conceptualized the relationship between integration and areas and
distances, you are ready to take a closer look at various applications;
these range from basic geometric identities to more advanced situations
in Physics and Engineering.

Unit 2 Time Advisory
This unit should take you 19.75 hours to complete.

☐ Subunit 2.1: 2.75 hours

☐ Subunit 2.2: 1 hour

☐ Subunit 2.3: 4.5 hours

☐ Sub-subunit 2.3.1: 3 hours

☐ Sub-subunit 2.3.2: 1.5 hours

☐ Subunit 2.4: 2.5 hours

☐ Subunit 2.5: 1.5 hours

☐ Subunit 2.6: 2 hours

☐ Subunit 2.7: 5.5 hours

☐ Sub-subunit 2.7.1: 1.5 hours

☐ Sub-subunit 2.7.2: 1.5 hours

☐ Sub-subunit 2.7.3: 0.5 hours

☐ Sub-subunit 2.7.4: 2 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Find the area between two curves.
- Find the volumes of solids using ideas from geometry.
- Find the volumes of solids of revolution using disks and washers.
- Find the volumes of solids of revolution using shells.
- Write and interpret a parameterization for a curve.
- Find the length of a curve.
- Find the surface area of a solid of revolution.
- Compute the average value of a function.
- Use integrals to compute the displacement and the total distance
traveled.
- Use integrals to compute moments and centers of mass.
- Use integrals to compute work.

2.1 The Area between CurvesSuppose you want to find the area between two concentric circles. How
would you do this? Logic dictates that you subtract the area of the
smaller circle from that of the larger circle. As this subunit will
demonstrate, this method also works when you are trying to determine the
area between curves.

Instructions: Please watch the segment of this video lecture from
time 21:30 minutes through the end. Note that lecture notes are
available in PDF; the link is on the same page as the lecture. In
this lecture, Dr. Jerison will explain how to calculate the area
between two curves.

Viewing this lecture and pausing to take notes should take
approximately 45 minutes.

Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “2. Applications of Integration,” and click
button 115 (Area between Curves I). Do problems 6-13. Next, choose
button 116 (Area between Curves II), and do problems 4-10. If at
any time a problem set seems too easy for you, feel free to move
on.

Completing these assessments should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Assessment: Indiana University Southeast: Margaret Ehringe’s
“Practice on Area between Two Curves”
Link: Indiana University Southeast: Margaret Ehringe’s “Section 5.3
Area between Two
Curves” (HTML)

Instructions: Click on the link above, and do problems 1-3 and
6-9. When you have finished, scroll down the page to check your
answers.

The point of this third assessment is for you to practice setting
up and completing these problems without the graphical aids provided
by the Temple University media; you will have to graph these curves
for yourself in order to begin the problems.

Completing this assessment should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

2.2 Volumes of SolidsWe often take basic geometric formulas for granted. (Have you ever
asked yourself why the volume of a right cylinder is
V=πhr2?) In this subunit, we will explore how some of these
formulas were developed. The key lies in viewing solids as functions
that revolve around certain lines. Consider, for example, a constant,
horizontal line, and then imagine that line revolving around the x-axis
(or any parallel line). The resulting shape is a right cylinder. We
can find the volume of this figure by looking at infinitesimally thin
“slices” and adding them all together. This concept enables us to
calculate the volume of some extremely complex figures. In this
subunit, we will learn how to do this in general; in the next, we will
now take a look at two conventional methods for doing so when the figure
has rotational symmetry.

Instructions: Please click on the link above, and work through each
of the three examples on the page. As in any assessment, solve the
problem on your own first. Solutions are given beneath each
example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the
kind permission of Elizabeth Wood, and can be viewed in its original
form
here (HTML).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.

2.3 Volume of Solids of RevolutionWhen we are presented with a solid that was produced by rotating a
curve around an axis, there are two sensible ways to take that solid
apart: slice it thinly perpendicularly to the axis, into disks (or
washers, if the solid had a hole in the middle), or peel layers from
around the outside like the paper wrapper of a crayon. The latter
method is known as the shell method and produces thin cylinders. In
both cases, we find the area of the thin segments and add them up to
find the volume; as usual, when we have infinitely many pieces, this
“addition” is really integration.

Instructions: Please click on the link above, and read Section 6.2
in its entirety (pages 308 through 318). This reading will cover
sub-subunits 2.3.1-2.3.2.
Studying this reading should take approximately 1 hour.
Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML). It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).

Instructions: Please click on the link above, and watch the
entirety of this video. Note that lecture notes are available in
PDF; the link is on the same page as the lecture. Dr. Jerison
elaborates on some tangential material for a few minutes in the
middle, but returns to the essential material very quickly. This
lecture will cover the topics outlined for sub-subunits 2.3.1 and
2.3.2.

Viewing this lecture and pausing to take notes should take
approximately 1 hour.

Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “2. Applications of Integration,” and click
button 119 (Solid of Revolution – Washers). Do problems 1-12. If
at any time a problem set seems too easy for you, feel free to move
on.

Completing this assessment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “2. Applications of Integration,” and click
button 120 (Solid of Revolution – Shells). Do problems 5-17. If at
any time a problem set seems too easy for you, feel free to move
on.
Completing this assessment should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

Instructions: This assessment is for subunits 2.2 and 2.3; do not
complete this assessment until you have worked through these
subunits in their entirety. Click on the link above, and work
through the exercises using the method you feel is most
appropriate.

2.4 Lengths of CurvesIn this subunit, we will make use of another concept that you have
known and understood for quite some time: the distance formula. If you
want to estimate the length of a curve on a certain interval, you can
simply calculate the distance between the initial point and terminal
point using the traditional formula. If you want to increase the
accuracy of this measurement, you can identify a third point in the
middle and calculate the sum of the two resulting distances. As we add
more points to the formula, our accuracy increases: the exact length of
the curve will be the sum (i.e. the integral) of the infinitesimally
small distances.

Instructions: Please click on the link above, and read Section 6.3
in its entirety (pages 319 through 325). This reading discusses how
to calculate the length of a curve, also known as arc length. This
includes calculating arc length for parametrically-defined curves.

Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “2. Applications of Integration,” and click
button 125 (Arc Length). Do all problems (1-9). If at any time a
problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

2.5 Surface Areas of SolidsIn this subunit, we will combine what we learned earlier in this unit.
Though you might expect that calculating the surface area of a solid
will be as easy as finding its volume, it actually requires a number of
additional steps. You will need to find the curve-length for each of
the “slices” we identified earlier and then add them together.

Reading: University of Wisconsin: H. Jerome Keisler’s Elementary
Calculus: Chapter 6: Applications of the Integral: “Section 6.4:
Area of a Surface of Revolution”
Link: University of Wisconsin: H. Jerome Keisler’s Elementary
Calculus: Chapter 6: Applications of the Integral: “Section 6.4:
Area of a Surface of
Revolution” (PDF)

Instructions: Please click on the link above, and read Section 6.4
in its entirety (pages 327 through 335). In this beautiful
presentation of areas of surfaces of revolution, the author again
makes use of rigorously-defined infinitesimals, as opposed to
limits. Recall that the approaches are equivalent; using an
infinitesimal is the same as using a variable and then taking the
limit as that variable tends to zero.

Instructions: Please click on the link above, and work through each
of the three examples on the page. As in any assessment, solve the
problem on your own first. Solutions are given beneath each
example.

Completing this assessment should take approximately 15-20
minutes.

Terms of Use: The linked material above has been reposted by the
kind permission of Elizabeth Wood, and can be viewed in its original
form
here (HTML).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.

2.6 Average Value of FunctionsNote: You probably learned about averages (or mean values) quite some
time ago. When you have a finite number of numerical values, you add
them together and divide by the number of values you have added. There
is nothing preventing us from seeking the average of an infinite number
of values (i.e. a function over a given interval). In fact, the formula
is intuitive: we add the numbers using an integral and divide by the
range.

Instructions: Please watch this video lecture from the beginning up
to time 30:00 minutes. Note that lecture notes are available in
PDF; the link is on the same page as the lecture. In this lecture,
Professor Jerison will explain how to calculate average values and
weighted average values.

Viewing this lecture and pausing to take notes should take
approximately 45 minutes.

Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “4. Assorted Application,” and click button
124 (Average Value). Do problems 3-11. If at any time a problem
set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

2.7 Physical ApplicationsWe will now apply what we have learned about integration to various
aspects of science. You may know that in physics, we calculate “work”
by multiplying the force of the work by the distance over which it is
exerted. You may also know that density is related to mass and volume.
But we now know that distance and volume are very much related to
integration. In this subunit, we will explore these and other
connections.

Instructions: Please click on the link above, and read the Section
9.2 in its entirety (pages 192 through 194).
Studying this reading should take approximately 15-20 minutes.
Terms of Use: The linked material above has been reposted by the
kind permission of David Guichard, and can be viewed in its original
form
[here](http://www.whitman.edu/mathematics/calculus/calculus_09_Applications_of_Integration.pdf#page=6)
(PDF). Please note that this material is under copyright and cannot
be reproduced in any capacity without explicit permission from the
copyright holder.

Instructions: Please click on the link above, and work through each
of the three examples on the page. As in any assessment, solve the
problem on your own first. Solutions are given beneath each
example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the
kind permission of Elizabeth Wood, and can be viewed in its original
form
here (HTML).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.

Studying this reading should take approximately 1 hour.
Terms of Use: The article above is released under a [Creative
Commons Attribution-NonCommercial-ShareAlike License
3.0](http://creativecommons.org/licenses/by-nc-sa/3.0/) (HTML). It
is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).

Also Available in:
[HTML](http://faculty.eicc.edu/bwood/math150supnotes/supplemental30.html)
Instructions: This assessment will test you on what you learned in
sub-subunits 2.7.2 and 2.7.3. Please click on the link above, and
work through each of the four examples on the page. As in any
assessment, solve the problem on your own first. Solutions are
given beneath each example.
Completing this assessment should take approximately 30 minutes.
Terms of Use: The material above has been reposted by the kind
permission of Elizabeth Wood, and can be viewed in its original
form [here](http://faculty.eicc.edu/bwood/math150supnotes/supplemental30.html) (HTML).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.

Instructions: Please click on the link above, and read Section 9.5
in its entirety (pages 205 through 208). Work is a fundamental
concept from physics roughly corresponding to the distance travelled
by an object multiplied by the force required to move it that
distance.
Studying this reading should take approximately 30 minutes.
Terms of Use: The linked material above has been reposted by the
kind permission of David Guichard, and can be viewed in its original
form
[here](http://www.whitman.edu/mathematics/calculus/calculus_09_Applications_of_Integration.pdf#page=20) (PDF).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.

Instructions: Please click on the link above, and work through each
of the seven examples on the page. As in any assessment, solve the
problem on your own first. Solutions are given beneath each
example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the
kind permission of Elizabeth Wood, and can be viewed in its original
form
here (HTML).
Please note that this material is under copyright and cannot be
reproduced in any capacity without explicit permission from the
copyright holder.